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Simplifying -7v(v + 3) + 4v + 5 = 7v + 9 Reorder the terms: -7v(3 + v) + 4v + 5 = 7v + 9 (3 * -7v + v * -7v) + 4v + 5 = 7v + 9 (-21v + -7v2) + 4v + 5 = 7v + 9 Reorder the terms: 5 + -21v + 4v + -7v2 = 7v + 9 Combine like terms: -21v + 4v = -17v 5 + -17v + -7v2 = 7v + 9 Reorder the terms: 5 + -17v + -7v2 = 9 + 7v Solving 5 + -17v + -7v2 = 9 + 7v Solving for variable 'v'. Reorder the terms: 5 + -9 + -17v + -7v + -7v2 = 9 + 7v + -9 + -7v Combine like terms: 5 + -9 = -4 -4 + -17v + -7v + -7v2 = 9 + 7v + -9 + -7v Combine like terms: -17v + -7v = -24v -4 + -24v + -7v2 = 9 + 7v + -9 + -7v Reorder the terms: -4 + -24v + -7v2 = 9 + -9 + 7v + -7v Combine like terms: 9 + -9 = 0 -4 + -24v + -7v2 = 0 + 7v + -7v -4 + -24v + -7v2 = 7v + -7v Combine like terms: 7v + -7v = 0 -4 + -24v + -7v2 = 0 Factor out the Greatest Common Factor (GCF), '-1'. -1(4 + 24v + 7v2) = 0 Ignore the factor -1.Subproblem 1
Set the factor '(4 + 24v + 7v2)' equal to zero and attempt to solve: Simplifying 4 + 24v + 7v2 = 0 Solving 4 + 24v + 7v2 = 0 Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. 0.5714285714 + 3.428571429v + v2 = 0 Move the constant term to the right: Add '-0.5714285714' to each side of the equation. 0.5714285714 + 3.428571429v + -0.5714285714 + v2 = 0 + -0.5714285714 Reorder the terms: 0.5714285714 + -0.5714285714 + 3.428571429v + v2 = 0 + -0.5714285714 Combine like terms: 0.5714285714 + -0.5714285714 = 0.0000000000 0.0000000000 + 3.428571429v + v2 = 0 + -0.5714285714 3.428571429v + v2 = 0 + -0.5714285714 Combine like terms: 0 + -0.5714285714 = -0.5714285714 3.428571429v + v2 = -0.5714285714 The v term is 3.428571429v. Take half its coefficient (1.714285715). Square it (2.938775513) and add it to both sides. Add '2.938775513' to each side of the equation. 3.428571429v + 2.938775513 + v2 = -0.5714285714 + 2.938775513 Reorder the terms: 2.938775513 + 3.428571429v + v2 = -0.5714285714 + 2.938775513 Combine like terms: -0.5714285714 + 2.938775513 = 2.3673469416 2.938775513 + 3.428571429v + v2 = 2.3673469416 Factor a perfect square on the left side: (v + 1.714285715)(v + 1.714285715) = 2.3673469416 Calculate the square root of the right side: 1.538618517 Break this problem into two subproblems by setting (v + 1.714285715) equal to 1.538618517 and -1.538618517.Subproblem 1
v + 1.714285715 = 1.538618517 Simplifying v + 1.714285715 = 1.538618517 Reorder the terms: 1.714285715 + v = 1.538618517 Solving 1.714285715 + v = 1.538618517 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.714285715' to each side of the equation. 1.714285715 + -1.714285715 + v = 1.538618517 + -1.714285715 Combine like terms: 1.714285715 + -1.714285715 = 0.000000000 0.000000000 + v = 1.538618517 + -1.714285715 v = 1.538618517 + -1.714285715 Combine like terms: 1.538618517 + -1.714285715 = -0.175667198 v = -0.175667198 Simplifying v = -0.175667198Subproblem 2
v + 1.714285715 = -1.538618517 Simplifying v + 1.714285715 = -1.538618517 Reorder the terms: 1.714285715 + v = -1.538618517 Solving 1.714285715 + v = -1.538618517 Solving for variable 'v'. Move all terms containing v to the left, all other terms to the right. Add '-1.714285715' to each side of the equation. 1.714285715 + -1.714285715 + v = -1.538618517 + -1.714285715 Combine like terms: 1.714285715 + -1.714285715 = 0.000000000 0.000000000 + v = -1.538618517 + -1.714285715 v = -1.538618517 + -1.714285715 Combine like terms: -1.538618517 + -1.714285715 = -3.252904232 v = -3.252904232 Simplifying v = -3.252904232Solution
The solution to the problem is based on the solutions from the subproblems. v = {-0.175667198, -3.252904232}Solution
v = {-0.175667198, -3.252904232}
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